At some point last night, it looks like Google changed something on their servers that broke the Gmail app on Android. Here are the details, as well as a workaround so you don't have to go email-less.
The issue has popped up as a Google Code bug, as well as on the Google Play page for Gmail and Android forums everywhere. Apparently, many users are experiencing an issue where Gmail can't open their primary account; instead, it just flickers and tries to reload the inbox without ever displaying it. Notifications still come through, and secondary accounts are still accessible, but primary accounts just won't work. There doesn't seem to be a common denominator that we can find, so we don't know who exactly is affected. It hasn't affected everyone, either (in fact, my Gmail is working just fine).
If you're having this problem, though, there is a simple thing you can do to get by until Google releases a fix. Just open up Android's generic Email app, log in with your Google credentials, and you should be able to access your inbox — no advanced IMAP settings required. You won't be able to archive emails or use any other Gmail-specific features, but at least you'll be able to read your email while we wait for Google to push out a fix. If you have this problem and know of another fix or workaround, be sure to let us know in the comments.